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IN THIS ISSUE

### Research Papers

J. Appl. Mech. 2008;75(2):021001-021001-11. doi:10.1115/1.2775493.

The motivation of this paper is to propose a methodology for analyzing the robust design optimization problem of complex dynamical systems excited by deterministic loads but taking into account model uncertainties and data uncertainties with an adapted nonparametric probabilistic approach, whereas only data uncertainties are generally considered in the literature by using a parametric probabilistic approach. The possible designs are represented by a numerical finite element model whose design parameters are deterministic and belong to an admissible set. The optimization problem is formulated for the stochastic system as the minimization of a cost function associated with the random response of the stochastic system including the variability of the stochastic system induced by uncertainties and the bias corresponding to the distance of the mean random response to a given target. The gradient and the Hessian of the cost function with respect to the design parameters are explicitly calculated. The complete theory and a numerical application are presented.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021002-021002-8. doi:10.1115/1.2775494.

A previous analytical solution of the deflection of a thin circular aeolotropic plate, with simply supported edge and uniform lateral load, has been used to derive approximate series expressions for the plate support reaction, which are directly applicable in practice. The support reaction, which has been calculated for some typical anisotropic materials of varying degree of anisotropy, varies significantly along the plate perimeter and strongly anisotropic materials require in general a higher order series solution. Certain solution constants of previous deflection approximations were not found to harmonize and are therefore recalculated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021003-021003-8. doi:10.1115/1.2775495.

Computing the angular velocity $ω$ from the angular acceleration matrix is a nonlinear problem that arises when one wants to estimate the three-dimensional angular velocity of a rigid-body from point-acceleration measurements. In this paper, two new methods are proposed, which compute estimates of the angular velocity from the symmetric part $WS$ of the angular acceleration matrix. The first method uses a change of coordinate frame of $WS$ prior to performing the square-root operations. The new coordinate frame is an optimal representation of $WS$ with respect to the overall error amplification. In the second method, the eigenvector spanning the null space of $WS$ is estimated. As $ω$ lies in this space, and because its magnitude is proportional to the absolute value of the trace of $WS$, it is a simple matter to obtain $ω$. A simulation shows that, for this example, the proposed methods are more accurate than those existing methods that use only centripetal acceleration measurements. Moreover, their errors are comparable to other existing methods that combine tangential and centripetal acceleration measurements. In addition, errors of 2.15% in the accelerometer measurements result in errors of approximately 3% in the angular-velocity estimates. This shows that accelerometers are competitive with angular-rate sensors for motions of the type of the simulated example, provided that position and orientation errors of the accelerometers are accounted for.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021004-021004-9. doi:10.1115/1.2775498.

In this paper, we present a new model to predict the fracture in brittle materials from a geometrical weakness presenting an arbitrary stress concentration. The main idea is to combine the strain gradient elasticity with a cohesive model that includes both the displacement and the rotation jumps between the cohesive surfaces in the separation law. Three material parameters were used in the establishment of the fracture criterion. The first two parameters are the commonly used $σc$, the ultimate stress, and $Gc$, the critical energy release rate. The third parameter is the characteristic length $l$ as in most of the strain gradient models. The proposed three-parameter model enables to take the different stress concentration levels into account, thus providing a criterion to predict fractures for any stress concentration, whether it is singular or not. Experimental results were selected to verify the accuracy and efficiency of the criterion. It was shown that the proposed model is physically reasonable, highly accurate, and easy to apply. It can be used in crack initiation prediction of engineering structures made of brittle materials.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021005-021005-11. doi:10.1115/1.2775499.

A new one-equation subgrid scale (SGS) model that makes use of the transport equation for the SGS kinetic energy $(kSGS)$ to calculate a representative velocity scale for the SGS fluid motion is proposed. In the $kSGS$ transport equation used, a novel approach is developed for the calculation of the rate of dissipation of the SGS kinetic energy $(ε)$. This new approach leads to an analytical computation of $ε$ via the assumption of a form for the energy spectrum. This introduces a more accurate representation of the dissipation term, which is then also used for the calculation of a representative length scale for the SGS based on their energy content. Therefore, the SG length scale is not associated simply with the grid resolution or the largest of the SGS but with a length scale representative of the overall SGS energy content. The formulation of the model is presented in detail, and the new approach is tested on a series of channel flow test cases with Reynolds number based on friction velocity varying from 180 to 1800. The model is compared with the Smagorinsky model (1963, “General Circulation Experiments With the Primitive Equations: 1. The Basic Experiment  ,” Mon. Weather Rev., 91, pp. 90–164) and the one-equation model of Yoshizawa and Horiuti (1985, “A Statistically-Derived Subgrid Scale Kinetic Energy Model for the Large Eddy Simulation of Turbulent Flows  ,” J. Phys. Soc. Jpn., 54(8), pp. 2834–2839). The results indicate that the proposed model can provide, on a given mesh, a more accurate representation of the SG scale effects.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021006-021006-7. doi:10.1115/1.2775501.

Numerical and analytical simulations of projectiles penetrating brittle materials such as ceramics and glasses are a very challenging problem. The difficulty comes from the fact that the yield surface of brittle materials is not well characterized (or even defined), and the failure process may change the material properties. Recently, some works have shown that it is possible to characterize and find the constitutive equation for brittle materials using a confined compression test, i.e., a test where a cylindrical specimen, surrounded by a confining sleeve, is being compressed axially by a mechanical testing machine. This paper focuses on understanding the confined compression test by presenting an analytical model that explicitly solves for the stresses and strains in the sample and the sleeve, assuming the sleeve is elastic and the specimen is elastoplastic with a Drucker–Prager plasticity model. The first part of the paper briefly explains the experimental technique and how the stress-strain curves obtained during the test are interpreted. A simple and straightforward approach to obtain the constitutive model of the material is then presented. Finally, a full analytical model with explicit solution for displacements, strains, and stresses in the specimen and the sleeve is described. The advantage of the analytical model is that it gives a full understanding of the test, as well as information that can be useful when designing the test (e.g., displacements of the outer radius of the specimen).

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021007-021007-8. doi:10.1115/1.2775504.

This paper considers a piezoelectric ceramic layer with a surface electrode. It focuses on the effect of the layer thickness on the electrode tip fields. A closed-form solution for the electromechanical fields at the electrode tip is obtained and is expressed in terms of the applied electric field intensity factor, which can be obtained exactly for infinite layer thickness and numerically for finite layer thickness. The stress, electric displacement, and electric field are plotted to show the effect of layer thickness. It is found that the stresses and field intensities at the electrode tip can be reduced considerably by decreasing the thickness of the piezoelectric layer, confirming the previous finding. The paper also gives a solution for two identical and collinear surface electrodes. The relative distance between the electrodes is observed to have significant influence on the electromechanical field in the piezoelectric layer.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021008-021008-8. doi:10.1115/1.2793130.

This paper outlines a multiscale model to quantitatively describe the chemomechanical coupling between adsorbed molecules and thin elastic films. The goal is to provide clear, quantitative connections between molecular interactions, adsorption distribution, and surface stress, which can be integrated with conventional thin film mechanics to quantify device performance in terms of molecular inputs. The decoupling of molecular and continuum frameworks enables a straightforward analysis of arbitrary structures and deformation modes, e.g., buckling and plate/membrane behavior. Moreover, it enables one to simultaneously identify both chemical properties (e.g., binding energy and grafting density) and mechanical properties (e.g., modulus and film geometry) that result in chemically responsive devices. We present the governing equations for scenarios where interactions between adsorbed molecules can be described in terms of pair interactions. These are used to quantify the mechanical driving forces that can be generated from adsorption of double-stranded DNA and $C60$ (fullerenes). The utility of the framework is illustrated by quantifying the performance of adsorption-driven cantilevers and clamped structures that experience buckling. We demonstrate that the use of surface-grafted polyelectrolytes (such as DNA) and ultracompliant elastomer structures is particularly attractive since deformation can be tuned over a very wide range by varying grafting density and chemical environment. The predictions illustrate that it is possible to construct (1) adsorption-based tools to quantify molecular properties such as polymer chain flexibility and (2) chemically activated structures to control flow in microfluidic devices.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021009-021009-9. doi:10.1115/1.2744036.

The paper presents the spectral stiffness microplane model, which is a general constitutive model for unidirectional composite laminates, able to simulate the orthotropic stiffness, prepeak nonlinearity, failure envelopes, and, in tandem with the material characteristic length, also the post-peak softening and fracture. The framework of the microplane model is adopted. The model exploits the spectral decomposition of the transversely isotropic stiffness matrix of the material to define orthogonal strain modes at the microplane level. This decomposition is a generalization of the volumetric-deviatoric split already used by Bažant and co-workers in microplane models for concrete, steel, rocks, soils, and stiff foams. Linear strain-dependent yield limits (boundaries) are used to provide bounds for the normal and tangential microplane stresses, separately for each mode. A simple version, with an independent boundary for each mode, can capture the salient aspects of the response of a unidirectional laminate, although a version with limited mode coupling can fit the test data slightly better. The calibration of model parameters, verification by test data, and analysis of multidirectional laminates are postponed for the subsequent companion paper.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021010-021010-6. doi:10.1115/1.2744037.

The spectral stiffness microplane (SSM) model developed in the preceding Part I of this study is verified by comparisons with experimental data for uniaxial and biaxial tests of unidirectional and multidirectional laminates. The model is calibrated by simulating the experimental data on failure stress envelopes analyzed in the recent so-called “World Wide Failure Exercise,” in which various existing theories were compared. The present theory fits the experiments as well as the theories that were best in the exercise. In addition, it can simulate the post-peak softening behavior and fracture, which is important for evaluating the energy-dissipation capability of composite laminate structures. The post-peak softening behavior and fracture are simulated by means of the crack band approach which involves a material characteristic length.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021011-021011-10. doi:10.1115/1.2793132.

This paper provides a simple, novel approach for synchronizing the motions of multiple “slave” nonlinear mechanical systems by actively controlling them so that they follow the motion of an independent “master” mechanical system. The multiple slave systems need not be identical to one another. The method is inspired by recent results in analytical dynamics, and it leads to the determination of the set of control forces to create such synchronization between highly nonlinear dynamical systems. No linearizations or approximations are involved, and the exact control forces needed to synchronize the nonlinear systems are obtained in closed form. The method is applied to the synchronization of multiple, yet different, chaotic gyroscopes that are required to replicate the motion of a master gyro, which may have a chaotic or a regular motion. The efficacy of the method and its simplicity in synchronizing these mechanical systems are illustrated by two numerical examples, the first dealing with a system of three different gyros, the second with five different ones.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021012-021012-9. doi:10.1115/1.2775496.

The moment Lyapunov exponents of a single degree-of-freedom viscoelastic system under the excitation of a wideband noise are studied in this paper. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The method of averaging, both first order and second order, is applied. The averaged Itô differential equation governing the $pth$ norm is established and the $pth$ moment Lyapunov exponent is then obtained. White noise and real noise are considered as models of wideband noises. The variations of the moment Lyapunov exponents with the change of different parameters are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021013-021013-9. doi:10.1115/1.2793133.

Free vibrations of a double-walled carbon nanotube (DWNT) are studied. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting each other by van der Waals forces. An original method is proposed to calculate the first seven order resonant frequencies and relative vibrational modes. Detailed results are demonstrated for DWNTs according to the different boundary conditions between inner and outer tubes, such as fixed-free, cantilever-free, fixed-simple and fixed-fixed (reduced form) supported ends. Our results indicate that there is a special invariable frequency for a DWNT that is not affected by different combinations of boundary conditions. All vibrational modes of the DWNT must be coaxial when the resonant frequency is smaller than this frequency. Some noncoaxial vibrations will occur when their resonant frequencies exceed the frequency. Especially, the first noncoaxial resonant frequency is still invariable for all different boundary conditions. A change of resonant frequency for various lengths of DWNTs is discussed in detail. In addition, our model predicts a new coaxial-noncoaxial vibrational mode in fixed-simple supports for inner and outer tubes of a DWNT.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021014-021014-7. doi:10.1115/1.2793134.

A novel moving force and prestress identification method based on finite element and wavelet-based method for bridge-vehicle system is developed. A two-axle vehicle model and simple-supported beam with prestressing force are studied. Finite element method is flexible in modeling structures with complex boundaries while the wavelet-analysis method has the characteristic of multiresolution and the ability to detect abrupt changes. Both methods are used in this work to identify the moving loads and prestressing force from the “measured” bridge responses, which may be strain or acceleration. Numerical simulations demonstrate the efficiency of the method under the effects of measurement noise, road roughness, sampling rate, and the arrangement of sensors with good accuracy. Results indicate that the proposed method has the advantages of both high computational performance and fine identification resolution.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021015-021015-8. doi:10.1115/1.2793135.

Aluminum shear panels can dissipate significant amount of energy through hysteresis provided strength deterioration due to buckling is avoided. A detailed experimental study of the inelastic behavior of the full-scale models of shear panels of 6063-O and 1100-O alloys of aluminum is conducted under slow cyclic loading of increasing displacement levels. The geometric parameters that determine buckling of the shear panels, such as web depth-to-thickness ratio, aspect ratio of panels, and number of panels, were varied among the specimens. Test results were used to predict the onset of buckling with proportionality factor $f$ in Gerard’s formulation of inelastic buckling. Moreover, a logarithmic relationship between buckling stress and slenderness ratio of the panel was observed to predict experimental data closely. These relations can be further used to determine the geometry of shear panels, which will limit the inelastic web buckling at design shear strains.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021016-021016-11. doi:10.1115/1.2793136.

Bifurcations of equilibria at bimodal branching points in potential systems are investigated. General formulas describing postbuckling paths and conditions for their stability are derived in terms of the original potential energy. Formulas describing unfolding of bimodal branching points due to a change of system parameters are given. A full list of possible cases for postbuckling paths, their stability, and unfolding depending on three system coefficients is presented. In order to calculate these coefficients, one needs the derivatives of the potential energy and eigenvectors of the linearized problem taken at the bifurcation point. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of an articulated elastic column having four degrees of freedom and depending on three problem parameters (stiffness coefficients at the hinges). For some of the bimodal critical points, numerical results are obtained illustrating influence of parameters on postbuckling paths, their stability, and unfolding. A surprising phenomenon that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode is recognized. An example with a constrained sum of the stiffnesses of the articulated column shows that the maximum critical load (optimal design) is attained at the bimodal point.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021017-021017-15. doi:10.1115/1.2793137.

This paper presents component-level empirical damage evolution regression models based on loads and damage information that do for mechanical damage prediction what the Paris law does for predicting crack growth under fatigue loading. Namely, these regression models combine information about the current damage state and internal system loads to predict the progress of damage to failure. One of the drawbacks of Paris-like crack evolution laws is that localized information about the loading (stress) and damage (crack length) is required. In structural health monitoring applications, it is not feasible to instrument every potential crack initiation region to collect this localized information. The component-level damage evolution regression models developed here only require global measurements that quantify the damage and loading at the level of the component rather than at the site of damage. This paper develops damage evolution regression models for an automotive sway bar link undergoing axial fatigue loading with two different damage mechanisms at a weldment and at an electrical discharge machining notch. Restoring force diagrams are used to calculate the load indicators as damage progresses and transmissibility functions are used to calculate the damage indicator during tests to failure. A component-level load intensity factor $(ΔK)$ is calculated during these tests so that the rate of damage accumulation can be used to predict the growth of damage and ultimate failure.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021018-021018-7. doi:10.1115/1.2793802.

Moving surface loads cause crack extension at a constant subcritical speed between perfectly bonded materials. The materials differ only in thermal properties and are governed by coupled thermoelastic equations that admit as special cases Fourier heat conduction and thermal relaxation with one or two relaxation times. Convection from the crack surfaces is allowed and for the latter two models is itself influenced by thermal relaxation. A dynamic steady state of plane strain is assumed. Fourier heat conduction is shown to dominate away from the crack edge at low speeds; solution behavior at the crack edge at high speeds depends upon the particular thermal model. Thermal mismatch is seen to cause solution behavior similar to that for the isothermal bimaterial, and so insight into the case of general material mismatch is provided.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021019-021019-12. doi:10.1115/1.2775502.

Various types of impinging jet flows are analytically modeled using inviscid free Gaussian jet solutions superimposed with experimentally fitted boundary layer models. Improved (more robust) and simplified solutions to existing models are defined. Velocity profiles, surface pressure distributions, and streamline plots are calculated for circular, plane, and annular impinging jets. The models show excellent agreement with existing experimental results in both laminar and turbulent conditions and for different Reynolds numbers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021020-021020-8. doi:10.1115/1.2775503.

The phenomenon of elastic boundary layers under quasistatic loading is investigated using the Floquet–Bloch formalism for two-dimensional, isotropic, periodic lattices. The elastic boundary layer is a region of localized elastic deformation, confined to the free edge of a lattice. Boundary layer phenomena in three isotropic lattice topologies are investigated: the semiregular Kagome lattice, the regular hexagonal lattice, and the regular fully triangulated lattice. The boundary layer depth is on the order of the strut length for the hexagonal and the fully triangulated lattices. For the Kagome lattice, the depth of boundary layer scales inversely with the relative density. Thus, the boundary layer in a Kagome lattice of low relative density spans many cells.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021021-021021-12. doi:10.1115/1.2755171.

Accumulative plastic deformation due to repeated loading is crucial to the lives of many mechanical components, such as gears, stamping dies, and rails in rail-wheel contacts. This paper presents a three-dimensional numerical model for simulating the repeated rolling or sliding contact of a rigid sphere over an elasto-plastic half-space. This model is a semi-analytical model based on the discrete convolution and fast Fourier transform algorithm. The half-space behaves either elastic-perfectly plastically or kinematic plastically. The analyses using this model result in histories of stress, strain, residual displacement, and plastic strain volume integral (PV) in the half-space. The model is examined through comparisons of the current results with those from the finite element method for a simple indentation test. The results of rolling contact obtained from four different hardening laws are presented when the load exceeds the theoretical shakedown limit. Shakedown and ratchetting behaviors are discussed in terms of the PV variation. The effect of friction coefficient on material responses to repeated sliding contacts is also investigated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):021022-021022-7. doi:10.1115/1.2755178.

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to uniform film stress and system curvature states over the entire system of a single thin film on a substrate. By considering a circular multilayer thin film/substrate system subjected to nonuniform temperature distributions, we derive relations between the stresses in each film and temperature, and between the system curvatures and temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part, which reflects the effect of temperature of other points on the location of scrutiny. We also derive relations between the film stresses in each film and the system curvatures, which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. The interfacial shear tractions between the films and between the film and substrate are proportional to the gradient of the first curvature invariant, and can also be inferred experimentally.

Commentary by Dr. Valentin Fuster

### Technical Briefs

J. Appl. Mech. 2008;75(2):024501-024501-3. doi:10.1115/1.2793131.

In prior work, Calsamiglia (1999, “Anomalous Frictional Behavior in Collisions of Thin Disks  ,” ASME J. Appl. Mech., 66, pp. 146–152) reported experimental results of collisions between thin plastic disks and a relatively rigid steel barrier. In those experiments, it was observed that, contrary to a commonly held assumption in rigid body collision modeling, the ratio of tangential to normal components of the contact impulse could be substantially less than the friction coefficient even for collisions where the disk contact point did not reverse its velocity direction (i.e., for sliding collisions). In those experiments, the disk’s edges were rounded to make the contact less sensitive to machining imperfections. While such impact/contact is nominally at a single point, the rounded edges make the interaction three dimensional (from the view point of analyzing deformations). Here, we revisit that problem computationally, but model the edges as flat, making the problem two dimensional. Our finite element calculations (ABAQUS ) do not reproduce the anomalous frictional interactions observed in those experiments, suggesting that rounding of the edges, among other possibilities, may have played a significant role in the experimental results.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):024502-024502-5. doi:10.1115/1.2775497.

A new rotating beam finite element is developed in which the basis functions are obtained by the exact solution of the governing static homogenous differential equation of a stiff string, which results from an approximation in the rotating beam equation. These shape functions depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. Using this new element and the Hermite cubic finite element, a convergence study of natural frequencies is performed, and it is found that the new element converges much more rapidly than the conventional Hermite cubic element for the first two modes at higher rotation speeds. The new element is also applied for uniform and tapered rotating beams to determine the natural frequencies, and the results compare very well with the published results given in the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2008;75(2):024503-024503-4. doi:10.1115/1.2775500.

We consider the dynamical response of a finite, simply supported Timoshenko beam loaded by a force moving with a constant velocity. The classical solution for the transverse displacement and the rotation of the cross section of a Timoshenko beam has a form of a sum of two infinite series, one of which represents the force vibrations (aperiodic vibrations) and the other one free vibrations of the beam. We show that one of the series, which represents aperiodic (force) vibrations of the beam, can be presented in a closed form. The closed form solutions take different forms depending if the velocity of the moving force is smaller or larger than the velocities of certain shear and bar velocities.

Commentary by Dr. Valentin Fuster

### Discussions

J. Appl. Mech. 2008;75(2):025501-025501-2. doi:10.1115/1.2775505.
FREE TO VIEW

The paper of McAdams (ASME J. Appl. Mech.74, pp. 191–202) explored two different approaches for damage detection in vibrating beams having both manufacturing variations in geometry and crack damage. One of the approaches, however, has a significant error in its formulation. The effects of this error on the formulation and the analytical results are discussed.

Commentary by Dr. Valentin Fuster